Lax Lean Archive

While this submission is a draft, it cannot be used by other submissions.

Symmetric low-rank factorization, including characteristic two

Lax342547.SymmetricFactorization · concepts/Lax342547/SymmetricFactorization.lean · lax-342547

proven

Loading review…

Community review

Flags

Each flag is tied to a public ORCID identity and explains why this concept may be incorrect.

No flags have been submitted.

    Community review

    Flag this concept

    State precisely what appears incorrect. This explanation will be public under your ORCID name.

    No source line selected.

    Natural Language Statement

    Theorem

    A symmetric rank-r matrix has a factorization Z H Zᵀ with Z injective and H symmetric and nonsingular. No assumption on the diagonal is made. Counting these factors gives the symmetric-matrix estimate in Lemma 5.5.

    Concept map
    3 concepts; 3 descendants hidden
    100%
    Proven claimDefinitionThis conceptRelated conceptA → B: B builds on A
    Evidence

    This concept declares 2 statements. Each proof establishes one of them relative to its assumptions.

    1 card_symmetric_rank proven

    2 symmetric_factorization proven

    Lean source view on GitHub

    1import Lax342547.LowRankCounting
    2
    3/-!
    4---
    5title: Symmetric low-rank factorization, including characteristic two
    6type: theorem
    7---
    8A symmetric rank-r matrix has a factorization Z H Zᵀ with Z injective
    9and H symmetric and nonsingular. No assumption on the diagonal is made.
    10Counting these factors gives the symmetric-matrix estimate in Lemma 5.5.
    11-/
    12
    13namespace Lax342547.SymmetricFactorization
    14
    15axiom symmetric_factorization {K I : Type} [Field K] [Fintype I]
    16 (M : Matrix I I K) (hM : M.transpose = M) :
    17 ∃ Z : Matrix I (Fin M.rank) K, ∃ H : Matrix (Fin M.rank) (Fin M.rank) K,
    18 Function.Injective Z.mulVec ∧ H.transpose = H ∧ Function.Bijective H.mulVec ∧
    19 Z * H * Z.transpose = M
    20
    21open Lax342547.MomentSpace
    22
    23axiom card_symmetric_rank {I : Type} [Fintype I] (r : ℕ) :
    24 Nat.card {M : Matrix I I Binary // M.transpose = M ∧ M.rank = r} ≤
    25 2 ^ (Fintype.card I * r + r * r)
    26
    27end Lax342547.SymmetricFactorization
    28
    Show ProofShow Proof
    Builds on
    Used by
    From Mathlib

    none

    Discussion

    Ask a question or add context. Endorsements and structured flags are kept in the review panel above.

    Loading discussion…